Solutions of the 3D inhomogeneous incompressible Navier-Stokes system with initial velocity in $VMO^{-1}$

Technical Summary: Solutions of the 3D Inhomogeneous Incompressible Navier-Stokes System with Initial Velocity in $VMO^{-1}$

Problem Statement
This paper investigates the initial value problem for the three-dimensional inhomogeneous incompressible Navier–Stokes equations:
$$\begin{cases} \partial_t \rho + u \cdot \nabla \rho = 0, \\ \partial_t (\rho u) + \text{div}(\rho u \otimes u) - \Delta u + \nabla P = 0, \\ \nabla \cdot u = 0, \\ (\rho, u)|_{t=0} = (\rho_0, u_0). \end{cases}$$
The system models mixtures of immiscible incompressible fluids with varying densities. The primary challenge addressed is the existence of strong solutions when the initial velocity $u_0$ possesses low regularity, specifically lying in the space $L^2 \cap VMO^{-1}$, and the initial density $\rho_0$ is merely bounded away from zero (with $C^1$ or $C^2$ regularity). The space $VMO^{-1}$ is a subspace of the critical space $BMO^{-1}$, defined as the closure of smooth functions with compact support in the $BMO^{-1}$ norm.

Methodology
The authors employ a "freezing coefficient" method adapted from [12] to handle the spatial inhomogeneity of the density. The core strategy involves the following steps:

1. Decomposition via Freezing Coefficients: For a fixed point $x_0$, the momentum equation is projected using the Leray operator $\mathbb{P}$, and the density coefficient $\rho$ is frozen at $\rho_0(x_0)$. This transforms the equation into a heat-type equation with variable coefficients. The solution $u$ is decomposed into a linear part ($u_L$), a bilinear part ($u_N$), and remainder terms ($u_R$) arising from the spatial variation of $\rho$ and the non-local nature of the pressure.
2. Functional Framework: The authors introduce a mixed-norm space $X_T$ to capture both critical and subcritical behaviors. The norm $\|u\|_{X_T}$ combines:
   • A critical component involving $L^\infty$ bounds weighted by $t^{1/2}$, $VMO^{-1}$-type norms ($V^b_T$), and Hölder continuity in time.
   • A subcritical component involving $L^{12}$ norms and higher-order time Hölder continuity.
     The inclusion of the $L^{12}$ norm is motivated by the fact that the Leray projection is not bounded on $L^\infty$ but is bounded on $L^p$ for $p \in (1, \infty)$.
3. Regularity of Density: Since the Lipschitz norm of the velocity gradient behaves like $t^{-1}$ (non-integrable), the transport equation for density leads to a loss of spatial regularity. The authors establish that if the initial density is in $C^1$ (or $C^2$), the solution retains specific Hölder regularity for small times, controlled by an auxiliary norm $\|u\|_T$.
4. Estimates for Remainder Terms: A significant technical contribution is the rigorous estimation of the remainder terms ($u_{R1}, u_{R2}, u_{R3}$). These terms involve the difference $\rho(y) - \rho(x_0)$. The authors utilize time-weighted Hölder norms and energy estimates (specifically $\sup t \|u_t\|_{L^2} < \infty$) to compensate for the loss of spatial regularity in the density. Lemma 4.1 provides a crucial tool for estimating time differences of integrals involving the heat kernel with variable coefficients.

Key Results

• Local Existence (Theorem 1.1):
  The authors prove the local existence of a strong solution for initial data $(\rho_0, u_0)$ where:

  • $\rho_0 \in C^1$ with a positive lower bound ($0 < C_1 < \rho_0 < C_2 < \infty$).
  • $u_0 \in L^2 \cap VMO^{-1}$.
  • The localized $BMO^{-1}$ norm $\|u_0\|_{BMO^{-1}_b}$ is sufficiently small ($\le \epsilon_0$).
    The solution exists on a time interval $[0, T]$ where $T \sim b^2$. The velocity satisfies specific bounds in the $X_T$ norm, and the density remains bounded.

• Global Existence (Theorem 1.2):
  Under the assumption that the initial density is close to a constant ($\rho_0 \in C^2$) and the initial velocity is sufficiently small in the $BMO^{-1}$ norm (specifically $\|\rho_0 - 1\|_{L^\infty} + \|u_0\|_{BMO^{-1}} < \epsilon_2$), the authors establish the global existence of a strong solution.

  • The proof relies on a "bootstrapping" argument: using the local existence result to show that the solution becomes sufficiently regular and small at a short time $T^*/2$ to satisfy the smallness condition required for global existence in the Besov space $\dot{B}^{-1+3/p}_{p,\infty}$ (referencing Theorem 6.1 from [25]).
  • The requirement for $\rho_0 \in C^2$ is necessary to ensure the maximal existence time in the local estimate does not depend on the smallness parameter in a way that prevents the global extension.

Significance and Claims
The paper claims to extend the well-posedness theory for inhomogeneous Navier–Stokes equations to the critical space $VMO^{-1}$ for the velocity, a setting that includes data with negative regularity.

• Novelty in Regularity: Unlike previous results requiring $u_0 \in L^p$ ($p>2$) or specific Besov spaces, this work accommodates $u_0 \in L^2 \cap VMO^{-1}$.
• Handling Inhomogeneity: The authors successfully adapt the freezing coefficient method to the inhomogeneous case, overcoming the difficulty that the density transport equation causes a loss of spatial regularity. They achieve this by exploiting time regularity and using subcritical norms to control the remainder terms.
• Uniqueness: The paper explicitly notes that establishing uniqueness under the assumptions of Theorem 1.1 (specifically for $u_0 \in L^2 \cap VMO^{-1}$ without smallness) remains an open problem. The methodology used for existence does not currently yield uniqueness in this critical setting, distinguishing it from the classical $BMO^{-1}$ results for homogeneous Navier–Stokes equations by Koch and Tataru [29].

Conclusion
This work provides a rigorous framework for solving the 3D inhomogeneous Navier–Stokes equations with low-regularity initial velocities in $VMO^{-1}$. By combining transport equation estimates with a refined freezing coefficient technique and a carefully constructed mixed-norm functional space, the authors establish local existence for general bounded densities and global existence when the density is close to a constant and the velocity is small. The results represent a significant step forward in understanding the well-posedness of fluid mixtures with variable densities in critical function spaces.